Math 1 · Unit 9 · Lesson 2

Arithmetic Sequences, Lines Made of Dots

An arithmetic sequence adds the same number each time, the common difference d. A recursive rule is the stairs: each term from the one before. An explicit rule is the elevator: straight to any term, a sub 1 plus d times n minus 1, because reaching the nth term takes n minus 1 steps. The common difference is the slope, and the y-intercept is the zeroth term, one step before the first.

About 45 minutes The common difference is the slope

The big idea

Adding gives arithmetic. An arithmetic sequence adds the same number, the common difference d, to get each next term, and you find d by subtracting any term minus the one before it. The recursive rule is the stairs: a sub 1 is the start, and a sub n is a sub n minus 1 plus d. The explicit rule is the elevator: a sub n is a sub 1 plus d times n minus 1, with n minus 1 because the first term is already there and you only take one step for each term after it. The dots sit on a line because the common difference is the slope. Simplify the explicit rule and you get slope-intercept form, and the y-intercept is the zeroth term, one step back from the first, so the first term is not the y-intercept. The explicit rule also runs backward: set it equal to a value and solve the Unit 1 way to find which term it is.

The parts of this lesson

Each part is its own short page. Go in order, or pick the one you need.

1The Words We UseThe vocabulary for this lesson, in one place.Start here 2Adding the same amount: the common differenceAdding gives arithmetic, multiplying gives geometric. 3It is a line: slope and the zeroth termThe dots sit on a line, so an arithmetic sequence must be a line in disguise. 4Running it backward: which term?The explicit rule also runs backward: given a term, find its term number. 5Practice ProblemsTry them, get help one step at a time, then print how it went.

Printables

One page with the common difference, both rules, the n minus 1 reason, the link to slope and the zeroth term, and a worked which-term problem. Plus a worksheet with an answer key and a short lesson plan.

Study guide (PDF) Worksheet and answer key (PDF) Lesson plan (PDF)

Where this goes next

Previous lesson: A Sequence Is a FunctionPosition in, term out. Next lesson: Geometric Sequences, Exponentials Made of DotsLists that multiply: the common ratio is the growth factor. Back to the Math 1 mapEvery Math 1 lesson, in course order. Tell us what you thoughtA few quick questions about this lesson. It takes a minute, and it is how the lesson gets better.

NC Math 1 standards in this lesson: NC.M1.F-IF.3 (terms of an arithmetic sequence come from a linear function), NC.M1.F-BF.1a (build linear functions, including arithmetic sequences), NC.M1.F-BF.2 (explicit and recursive forms).